**2. Stirred bioreactor**

In this case study, some of the methodological procedures described in part one of this series are used to numerically simulate three different stirred bioreactors to determine the specific power input and the *k*L*a* value. The specific power input was determined for two different stirrers (stirrer diameter of 20 mm and 40 mm) according to Bauer et al. [1] for a bearing-free, magnetically driven 2 L benchtop system described by Schirmer et al. [2], based on Levitronix's levitating impeller technology using the BPS-i30 drive (**Figure 1**) to validate the simulations described below. For this purpose, the torque was calculated using the known motor constant *K*<sup>t</sup> with 1*:*13 N cm A�<sup>1</sup> using eqs (1) and (2) and the required current for stirring:

$$M = K\_\mathbf{t} \cdot I \tag{1}$$

$$P/V = \frac{2 \cdot \pi \cdot n \cdot M}{V} \tag{2}$$

To determine the specific power input using a numerical simulation, the power input of the stirrer was determined based on the predicted fluid flow (see **Figure 1**) and the acting torque. For this purpose, Ansys Fluent was used with the realisable *k*-*ε*-model [3]. Both the vessel walls and the impeller were treated as nonslip boundaries with standard wall functions and the axial velocity at the fluid surface was set to zero. The stirrer rotation was implemented using the multiple reference frame (MRF) method. Discretisation was performed using the first-order upwind scheme and pressure-velocity coupling using the COUPLED algorithm. The fluid domain was defined by an unstructured mesh of approximately 8 � <sup>10</sup><sup>6</sup> tetrahedrons. The specific power input for the working volumes of 2 L and 4 L was determined for the bacterial

*Computational Fluid Dynamics for Advanced Characterisation of Bioreactors Used… DOI: http://dx.doi.org/10.5772/intechopen.109849*

**Figure 1.**

*Computer-aided design (CAD) geometry of the setup and design of the bearing-free, magnetically driven 2 L benchtop system based on Levitronix's levitating impeller technology (left) and numerically derived fluid flow field with a stirrer diameter of 40 mm (right).*

version of the Minifors 2 (Infors AG) with a total volume of 6 L, using OpenFOAM's simpleFoam algorithm, the Gauss upwind scheme, the MRF method, and the *k*-*ε*model of Launder and Spalding [4]. Discretisation of the fluid domain was performed with 1*:*<sup>7</sup> 106 cells (2 L) and 3 <sup>10</sup><sup>6</sup> cells (4 L), respectively. Validation of the numerical simulations was performed by determining the specific power input at the same working volumes using a torque meter to which the shaft and the stirrers were attached [1, 5, 6].

The results illustrated in **Figure 2** show good correlation between the experimentally and numerically determined specific power inputs for both the magnetically driven system with the BPS-i30 drive and the Minifors 2. With the exception of the 40-mm diameter stirrer in combination with the BPS-i30, all the other configurations resulted in values >5 kW m3, making them suitable for microbial applications.

In contrast to the previous ones shown, which were single phase and steady state, the oxygen transfer in another 2 L stirred bioreactor (modified HyPerforma glass bioreactor from Thermo Fisher Scientific Inc.) was determined transiently via a twophase Euler-Euler simulation [7] (**Figure 3** left). Besides the rotation of the stirrer which was realised via MRF, the aeration was defined by a gas inlet boundary condition at the sparger (a gas outlet is defined at the bioreactor lid). The liquid side mass transfer coefficient *k*<sup>L</sup> value can be calculated according to eq. (3) as a function of the energy dissipation rate (Brüning's [8] adapted formulation was used for these studies). The specific interface *a* can be calculated according to eq. (4) [9]. Thus, the volumetric oxygen mass transfer coefficient *k*L*a* can be determined, by the product of *k*<sup>L</sup> and *a*. The class method was used to model the gas bubbles and their size distribution with the population balance modelling approach in this example. 24 gas bubble size classes were used, which according to our own investigations and literature values proved to be a good compromise between accuracy and computation time (the computation time increases exponentially as the class number increases) [7, 10, 11]. The model of Laakkonen et al. [12] with binary bubble breakup was used to model bubble

#### **Figure 2.**

*Double logarithmic representation of the numerically and experimentally determined specific power inputs of the magnetically driven system using the BPS-i30 drive with stirrer diameters of 20 mm and 40 mm as well as of the bacterial version of the Minifors 2 with working volumes of 2 L and 4 L.*

#### **Figure 3.**

*k*L*a value determination by means of CFD coupled with population balance modelling. CFD simulation of the aerated and stirred 2 L HyPerforma glass bioreactor (left). The colour map corresponds to the logarithmised relative gas fraction. The gas flow is visualised using line integral convolution. Temporal variability of the k*L*a value determined by CFD simulation and its validation (right). Since the gassing-out method was used for the validation, no temporal statement about the k*L*a value is available.*

*Computational Fluid Dynamics for Advanced Characterisation of Bioreactors Used… DOI: http://dx.doi.org/10.5772/intechopen.109849*

breakup, and the model of Coulaloglou and Tavlarides [13] was used to model the coalescence behaviour. A review of the influence of different coalescence and breakup models can be found in Seidel and Eibl [7]. The interfacial models also have a significant influence on the predicted specific surface area. Different models were investigated by Seidel and Eibl [7], with the model of Schiller and Naumann [14] proving to be the best for calculating the drag force, the model of Tomiyama et al. [15] for calculating the lift force, and the model of Lamb [16] for calculating the virtual mass force. Using this model, a *<sup>k</sup>*L*<sup>a</sup>* value of 14 ð Þ *:*<sup>3</sup> � <sup>0</sup>*:*<sup>6</sup> <sup>h</sup>�<sup>1</sup> was determined for a stirrer speed of 600 rpm and an aeration rate of 0.5 vvm (**Figure 3** right). This result was verified experimentally using the gassing-out method according to the DECHEMA e.V. Working Group for Single-Use Technology [1, 17]. An eightfold determination resulted in a measured value of 11 ð Þ *:*<sup>1</sup> � <sup>0</sup>*:*<sup>2</sup> <sup>h</sup>�<sup>1</sup> . This deviation can be explained not only by the discretisation error but also much more by the modelling error. The use of various models, some of which are based on empirical approximations, can lead to deviations from the experimental investigations. Only the MRF approach was chosen for the stirrer motion, which is less accurate than the dynamic mesh approach. Nevertheless, this method is suitable for estimating the *k*L*a* value in stirred systems:

$$k\_{\rm L} = \frac{2}{\sqrt{\pi}} \epsilon \nu \left(\frac{D\_{O\_2}}{\nu}\right)^{\frac{1}{4}} \tag{3}$$

$$a = \frac{A\_{\oplus}}{V} = \frac{\mathfrak{G}a}{d\_{\mathfrak{A}}} \tag{4}$$
